Save Rooms (or Save Points) are safe and holy areas within Dracula's Castle (or other locations) where a weary traveler can rest and restore his or her health. Enemies cannot appear inside a save room. These rooms became common areas within the Castlevania series once the Metroid theme-styled Symphony of the Night was released in 1997. As such, these locations allow the player to do both, restore HP and MP, and also save the game's progress. They won't restore used Hearts, though.
A save room usually features the statue of a goddess or an angel who heals the player. In Symphony of the Night, a polyhedron can be found spinning in midair instead; this transforms into a golden coffin, wrapping around Alucard for him to sleep inside. In that same game, Alucard manages to find a fake save room where he's attacked in his dreams by a Succubus (it should be noted that this coffin is purple colored).
Save rooms are common and numerous throughout Castlevania and were implemented due to newer gaming systems no longer needing a password feature.